We investigate the regular and chaotic motions of a single Paul-trapped ion under the pseudopotential approximation, which interacts with a double-δ-kicked periodic potential. We apply the method of integral-equation to construct the exact solution of the classical equation and use the numerical technique to plot the orbits in phase space and the time evolutions of average energies. Combining the analytical results with the numerical ones, we arrived at two interesting conclusions: Firstly, there are regular stable motions of the double δ-kicked system,at variance with the resonance case of the corresponding single δ-Kicked system. Secondly, when the time interval between the double δ pulses becomes shorter, the regular motion of the system becomes chaotic, and the speed of classical diffusion of the average energy is related to the degree of chaos. It is shown that the resonance may lead to loss of stability and the instability can be controlled by adjusting the laser wave vector.
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